Model

Literal listener (tries to infer the probability \(\phi\) of getting a blue gumball given an utterance \(u\)):

\[P_L(\phi \mid u; \theta) \propto P(\phi) \left( {0.95} \times \mathbb{1}[\phi > \theta_{u}] + 0.05 * P_{uniform}(\phi; 0,1) \right)\] (The term \(0.05 * P_{uniform}(\phi; 0,1)\)) corresponds to the noise term which assigns a small non-zero probability to all \(\phi\) idependent of the actual utterance.) We assume that the prior \(P(\phi)\) is uniform.

Pragmatic speaker (marginalizes over all possible values of \(\theta\)):

\[P_S(u \mid \phi, condition) \propto \int P(\theta) \exp\left(\lambda * \left(\log P_L(\phi \mid u; \theta) - c(u, condition)\right)\right) d\theta \]

Costs:

\[ c(u, condition) = \begin{cases} 0 &\quad\text{if } u \text{ is one of the utterances in } condition\\ c_{u} &\quad\text{otherwise} \\ \end{cases} \]

Prior over thresholds \(\theta\):

\[P(\theta_u) = Beta(\alpha_u, \beta_u)\]

Estimated parameters:

\(\alpha_u, \beta_u \sim Uniform(0,30)\), \(c_u\sim Uniform(0,5)\), \(\lambda \sim Uniform(.1,3)\)

Left column: Experimental data.

Middle column: Model predictions with parameters estimated from all conditions.

Right column: Model predictions with parameters estimated from all conditions but the current one.

Experimental data and model predictions

## R^2: 0.962100734845317 
## R^2: 0.901785871999164

## R^2: 0.893958743002193 
## R^2: 0.814469067400412

## R^2: 0.92887618625227 
## R^2: 0.816326863527004

## R^2: 0.911223847778544 
## R^2: 0.820477601988968

## R^2: 0.92681575326815 
## R^2: 0.870527183689948

## R^2: 0.884452278220882 
## R^2: 0.853405850344422

## R^2: 0.803185003839032 
## R^2: 0.697707034566082

## R^2: 0.885954531882677 
## R^2: 0.881768117805778

## R^2: 0.916130676480786 
## R^2: 0.877186630997309

## R^2: 0.89054350714818 
## R^2: 0.871770190996445

## R^2: 0.831921319476047 
## R^2: 0.805277975525553

## R^2: 0.891647987894687 
## R^2: 0.863670019341647

## R^2: 0.853554107925385 
## R^2: 0.814132218781507

## R^2: 0.822272345719847 
## R^2: 0.795077965064735

## R^2: 0.891972040221257 
## R^2: 0.871765228009494

## R^2: 0.848189269538628 
## R^2: 0.855358423129167

Threshold distributions